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Chapter 5: Problem 94

Simplify. $$ 3\left(2 y^{2}-y-1\right)-\left(6 y^{2}-3 y-3\right) $$

Short Answer

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Step by step solution

01

- Distribute the Multiplication

Distribute the 3 to each term inside the first parenthesis: 3(2y^2 - y - 1) = 6y^2 - 3y - 3.
02

- Distribute the Negative Sign

Distribute the negative sign to each term inside the second parenthesis: -(6y^2 - 3y - 3) = -6y^2 + 3y + 3.
03

- Combine Like Terms

Combine the like terms from both expressions: 6y^2 - 3y - 3 - 6y^2 + 3y + 3.
04

- Simplify the Expression

Cancel out terms that add up to zero: 6y^2 - 6y^2 + (-3y + 3y) + (-3 + 3) = 0. So, the simplified expression is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental algebraic principle. It allows you to multiply a single term by each term within a parenthesis. This property helps us break down complex expressions.
If you have an expression like \(a(b + c)\), this becomes \(ab + ac\).

In our exercise, we use this to simplify \(3(2y^2 - y - 1)\). We distribute the \(3\) to each term inside the parenthesis:
\[3 \times 2y^2 + 3 \times (-y) + 3 \times (-1) = 6y^2 - 3y - 3\]
This is the first step to making the expression easier to manage.
Negative Sign Distribution
Distributing a negative sign can sometimes trip students up. It is similar to distributing a number, but you must change the sign of each term.

In our example, we have \(-(6y^2 - 3y - 3)\). We distribute the negative sign to each term inside the parenthesis:
\[-1 \times 6y^2 + -1 \times (-3y) + -1 \times (-3) = -6y^2 + 3y + 3\]

Each term inside the parenthesis changes its sign: positive becomes negative and vice versa. This step is crucial for correctly simplifying algebraic expressions.
Combining Like Terms
Combining like terms means adding or subtracting terms that have the same variable and exponent.

In our step, we look at: \(6y^2 - 3y - 3 - 6y^2 + 3y + 3\). We identify and combine like terms:
  • \(6y^2\) and \(-6y^2\)
  • \(-3y\) and \(3y\)
  • \(-3\) and \(3\)


When we combine them, we get:
\(6y^2 - 6y^2 + (-3y + 3y) + (-3 + 3)\). These terms effectively cancel each other out, leading to a simplified expression of \(0\).
Simplification
Simplification is the process of reducing an expression to its most basic form. This makes it much easier to understand and use in further calculations.

In this problem, after distributing and combining like terms, we're left with:
\(6y^2 - 6y^2 + (-3y + 3y) + (-3 + 3)\)
Each pair of terms adds up to zero:
  • \(6y^2 - 6y^2 = 0\)
  • \(-3y + 3y = 0\)
  • \(-3 + 3 = 0\)

Thus, the entire expression simplifies down to \(0\). When simplifying, always look to cancel out terms where possible. This leads to the most reduced form of the expressions.

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